The existence of Euler and Hamiltonian graph make it easier to solve a real-life problem. During the time of pandemic “Covid-19”, it is very essential for each one of us to be vaccinated. Vaccination is done in the hospitals by using Eulerian and Hamiltonian graphs not only to prevent people from infecting but also to increase the speed of vaccination. In this paper, the authors have discussed the use of graph theory to effectively handle the covid – 19 pandemic situations in the allocation of patients to appropriate hospitals. The work done in this paper may be utilized for the hospital–patient management as well as for the study by upcoming researchers to utilize the graph theory in any complicated situations.